Lynn Marecek; MaryAnne Anthony-Smith

Review Exercises

Simplify and Use Square Roots

Simplify Expressions with Square Roots

In the following exercises, simplify.

606.

$\sqrt{64}$

607.

$\sqrt{144}$

608.

$- \sqrt{25}$

609.

$- \sqrt{81}$

610.

$\sqrt{−9}$

611.

$\sqrt{−36}$

612.

$\sqrt{64} + \sqrt{225}$

613.

$\sqrt{64 + 225}$

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

614.

$\sqrt{28}$

615.

$\sqrt{155}$

Approximate Square Roots

In the following exercises, approximate each square root and round to two decimal places.

616.

$\sqrt{15}$

617.

$\sqrt{57}$

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

618.

$\sqrt{q^{2}}$

619.

$\sqrt{64 b^{2}}$

620.

$− \sqrt{121 a^{2}}$

621.

$\sqrt{225 m^{2} n^{2}}$

622.

$− \sqrt{100 q^{2}}$

623.

$\sqrt{49 y^{2}}$

624.

$\sqrt{4 a^{2} b^{2}}$

625.

$\sqrt{121 c^{2} d^{2}}$

Simplify Square Roots

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

626.

$\sqrt{300}$

627.

$\sqrt{98}$

628.

$\sqrt{x^{13}}$

629.

$\sqrt{y^{19}}$

630.

$\sqrt{16 m^{4}}$

631.

$\sqrt{36 n^{13}}$

632.

$\sqrt{288 m^{21}}$

633.

$\sqrt{150 n^{7}}$

634.

$\sqrt{48 r^{5} s^{4}}$

635.

$\sqrt{108 r^{5} s^{3}}$

636.

$\frac{10 - \sqrt{50}}{5}$

637.

$\frac{6 + \sqrt{72}}{6}$

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

638.

$\sqrt{\frac{16}{25}}$

639.

$\sqrt{\frac{81}{36}}$

640.

$\sqrt{\frac{x^{8}}{x^{4}}}$

641.

$\sqrt{\frac{y^{6}}{y^{2}}}$

642.

$\sqrt{\frac{98 p^{6}}{2 p^{2}}}$

643.

$\sqrt{\frac{72 q^{8}}{2 q^{4}}}$

644.

$\sqrt{\frac{65}{121}}$

645.

$\sqrt{\frac{26}{169}}$

646.

$\sqrt{\frac{64 x^{4}}{25 x^{2}}}$

647.

$\sqrt{\frac{36 r^{10}}{16 r^{5}}}$

648.

$\sqrt{\frac{48 p^{3} q^{5}}{27 p q}}$

649.

$\sqrt{\frac{12 r^{5} s^{7}}{75 r^{2} s}}$

Add and Subtract Square Roots

Add and Subtract Like Square Roots

In the following exercises, simplify.

650.

$3 \sqrt{2} + \sqrt{2}$

651.

$5 \sqrt{5} + 7 \sqrt{5}$

652.

$4 \sqrt{y} + 4 \sqrt{y}$

653.

$6 \sqrt{m} - 2 \sqrt{m}$

654.

$−3 \sqrt{7} + 2 \sqrt{7} - \sqrt{7}$

655.

$8 \sqrt{13} + 2 \sqrt{3} + 3 \sqrt{13}$

656.

$3 \sqrt{5 x y} - \sqrt{5 x y} + 3 \sqrt{5 x y}$

657.

$2 \sqrt{3 r s} + \sqrt{3 r s} - 5 \sqrt{r s}$

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

658.

$\sqrt{32} + 3 \sqrt{2}$

659.

$\sqrt{8} + 3 \sqrt{2}$

660.

$\sqrt{72} + \sqrt{50}$

661.

$\sqrt{48} + \sqrt{75}$

662.

$3 \sqrt{32} + \sqrt{98}$

663.

$\frac{1}{3} \sqrt{27} - \frac{1}{8} \sqrt{192}$

664.

$\sqrt{50 y^{5}} - \sqrt{72 y^{5}}$

665.

$6 \sqrt{18 n^{4}} - 3 \sqrt{8 n^{4}} + n^{2} \sqrt{50}$

Multiply Square Roots

Multiply Square Roots

In the following exercises, simplify.

666.

$\sqrt{2} \cdot \sqrt{20}$

667.

$2 \sqrt{2} \cdot 6 \sqrt{14}$

668.

$\sqrt{2 m^{2}} \cdot \sqrt{20 m^{4}}$

669.

$(6 \sqrt{2 y}) (3 \sqrt{50 y^{3}})$

670.

$(6 \sqrt{3 v^{4}}) (5 \sqrt{30 v})$

671.

${(\sqrt{8})}^{2}$

672.

${(− \sqrt{10})}^{2}$

673.

$(2 \sqrt{5}) (5 \sqrt{5})$

674.

$(−3 \sqrt{3}) (5 \sqrt{18})$

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

675.

$10 (2 - \sqrt{7})$

676.

$\sqrt{3} (4 + \sqrt{12})$

677.

$(5 + \sqrt{2}) (3 - \sqrt{2})$

678.

$(5 - 3 \sqrt{7}) (1 - 2 \sqrt{7})$

679.

$(1 - 3 \sqrt{x}) (5 + 2 \sqrt{x})$

680.

$(3 + 4 \sqrt{y}) (10 - \sqrt{y})$

681.

${(1 + 6 \sqrt{p})}^{2}$

682.

${(2 - 6 \sqrt{5})}^{2}$

683.

$(3 + 2 \sqrt{7}) (3 - 2 \sqrt{7})$

684.

$(6 - \sqrt{11}) (6 + \sqrt{11})$

Divide Square Roots

Divide Square Roots

In the following exercises, simplify.

685.

$\frac{\sqrt{75}}{10}$

686.

$\frac{2 - \sqrt{12}}{6}$

687.

$\frac{\sqrt{48}}{\sqrt{27}}$

688.

$\frac{\sqrt{75 x^{7}}}{\sqrt{3 x^{3}}}$

689.

$\frac{\sqrt{20 y^{5}}}{\sqrt{2 y}}$

690.

$\frac{\sqrt{98 p^{6} q^{4}}}{\sqrt{2 p^{4} q^{8}}}$

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

691.

$\frac{10}{\sqrt{15}}$

692.

$\frac{6}{\sqrt{6}}$

693.

$\frac{5}{3 \sqrt{5}}$

694.

$\frac{10}{2 \sqrt{6}}$

695.

$\sqrt{\frac{3}{28}}$

696.

$\sqrt{\frac{9}{75}}$

Rationalize a Two Term Denominator

In the following exercises, rationalize the denominator.

697.

$\frac{4}{4 + \sqrt{27}}$

698.

$\frac{5}{2 - \sqrt{10}}$

699.

$\frac{4}{2 - \sqrt{5}}$

700.

$\frac{5}{4 - \sqrt{8}}$

701.

$\frac{\sqrt{2}}{\sqrt{p} + \sqrt{3}}$

702.

$\frac{\sqrt{x} - \sqrt{2}}{\sqrt{x} + \sqrt{2}}$

Solve Equations with Square Roots

Solve Radical Equations

In the following exercises, solve the equation.

703.

$\sqrt{7 z + 1} = 6$

704.

$\sqrt{4 u - 2} - 4 = 0$

705.

$\sqrt{6 m + 4} - 5 = 0$

706.

$\sqrt{2 u - 3} + 2 = 0$

707.

$\sqrt{u - 4} + 4 = u$

708.

$\sqrt{v - 9} + 9 = 0$

709.

$\sqrt{r - 4} - r = −10$

710.

$\sqrt{s - 9} - s = −9$

711.

$2 \sqrt{2 x - 7} - 4 = 8$

712.

$\sqrt{2 - x} = \sqrt{2 x - 7}$

713.

$\sqrt{a} + 3 = \sqrt{a + 9}$

714.

$\sqrt{r} + 3 = \sqrt{r + 4}$

715.

$\sqrt{u} + 2 = \sqrt{u + 5}$

716.

$\sqrt{n + 11} - 1 = \sqrt{n + 4}$

717.

$\sqrt{y + 5} + 1 = \sqrt{2 y + 3}$

Use Square Roots in Applications

In the following exercises, solve. Round approximations to one decimal place.

718.

A pallet of sod will cover an area of about 600 square feet. Trinh wants to order a pallet of sod to make a square lawn in his backyard. Use the formula $s = \sqrt{A}$ to find the length of each side of his lawn.

719.

A helicopter dropped a package from a height of 900 feet above a stranded hiker. Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the hiker.

720.

Officer Morales measured the skid marks of one of the cars involved in an accident. The length of the skid marks was 245 feet. Use the formula $s = \sqrt{24 d}$ to find the speed of the car before the brakes were applied.

Higher Roots

Simplify Expressions with Higher Roots

In the following exercises, simplify.

721.

ⓐ $\sqrt[6]{64}$
ⓑ $\sqrt[3]{64}$

722.

ⓐ $\sqrt[3]{−27}$
ⓑ $\sqrt[4]{−64}$

723.

ⓐ $\sqrt[9]{d^{9}}$
ⓑ $\sqrt[8]{v^{8}}$

724.

ⓐ $\sqrt[5]{a^{10}}$
ⓑ $\sqrt[3]{b^{27}}$

725.

ⓐ $\sqrt[4]{16 x^{8}}$
ⓑ $\sqrt[6]{64 y^{12}}$

726.

ⓐ $\sqrt[7]{128 r^{14}}$
ⓑ $\sqrt[4]{81 s^{24}}$

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

727.

ⓐ $\sqrt[9]{d^{9}}$
ⓑ

728.

ⓐ $\sqrt[3]{54}$
ⓑ $\sqrt[4]{128}$

729.

ⓐ $\sqrt[5]{64 c^{8}}$
ⓑ $\sqrt[4]{48 d^{7}}$

730.

ⓐ $\sqrt[3]{343 q^{7}}$
ⓑ $\sqrt[6]{192 r^{9}}$

731.

ⓐ $\sqrt[3]{−500}$
ⓑ $\sqrt[4]{−16}$

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

732.

$\sqrt[5]{\frac{r^{10}}{r^{5}}}$

733.

$\sqrt[3]{\frac{w^{12}}{w^{2}}}$

734.

$\sqrt[4]{\frac{64 y^{8}}{4 y^{5}}}$

735.

$\sqrt[3]{\frac{54 z^{9}}{2 z^{3}}}$

736.

$\sqrt[6]{\frac{64 a^{7}}{b^{2}}}$

Add and Subtract Higher Roots

In the following exercises, simplify.

737.

$4 \sqrt[5]{20} - 2 \sqrt[5]{20}$

738.

$4 \sqrt[3]{18} + 3 \sqrt[3]{18}$

739.

$\sqrt[4]{1250} - \sqrt[4]{162}$

740.

$\sqrt[3]{640 c^{5}} - \sqrt[3]{−80 c^{3}}$

741.

$\sqrt[5]{96 t^{8}} + \sqrt[5]{486 t^{4}}$

Rational Exponents

Simplify Expressions with $a^{\frac{1}{n}}$

In the following exercises, write as a radical expression.

742.

$r^{\frac{1}{8}}$

743.

$s^{\frac{1}{10}}$

In the following exercises, write with a rational exponent.

744.

$\sqrt[5]{u}$

745.

$\sqrt[6]{v}$

746.

$\sqrt[3]{9 m}$

747.

$\sqrt[6]{10 z}$

In the following exercises, simplify.

748.

$16^{\frac{1}{4}}$

749.

$32^{\frac{1}{5}}$

750.

${(−125)}^{\frac{1}{3}}$

751.

${(125)}^{- \frac{1}{3}}$

752.

${(−9)}^{\frac{1}{2}}$

753.

${(36)}^{- \frac{1}{2}}$

Simplify Expressions with $a^{\frac{m}{n}}$

In the following exercises, write with a rational exponent.

754.

$\sqrt[3]{q^{5}}$

755.

$\sqrt[5]{n^{8}}$

In the following exercises, simplify.

756.

$27^{- \frac{2}{3}}$

757.

$64^{\frac{5}{2}}$

758.

$36^{\frac{3}{2}}$

759.

$81^{- \frac{5}{2}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

760.

$3^{\frac{4}{5}} \cdot 3^{\frac{6}{5}}$

761.

${(x^{6})}^{\frac{4}{3}}$

762.

$\frac{z^{\frac{5}{2}}}{z^{\frac{7}{5}}}$

763.

${(16 s^{\frac{9}{4}})}^{\frac{1}{4}}$

764.

${(m^{8} n^{12})}^{\frac{1}{4}}$

765.

$\frac{z^{\frac{2}{3}} \cdot z^{- \frac{1}{3}}}{z^{- \frac{5}{3}}}$